XYZ Ltd produces three products $,$X Y and Z$,$The capacity of XYZ Ltd$\text{'}$s plant is restricted by process alpha. Process alpha is expected to be operational for eighty hours per day and can produce $1200$ units of X per hour,$1500$ units of Y per hour and $600$ units of Z per hour. Selling prices and material costs for each product are as follows. Product X k$150$ per unit material cost k$80$ per unit, Product Y selling price k$120$ per unit Material cost k$40$ per unit, Product Z selling price k$300$ per unit Material cost k$100$ per unit. Conversion costs are k$720,000$ per day.

Required: $$

$[$a$]$ calculate the profit per day if daily output achieved is $6000$ units of X$,4500$ units of Y and $1200$ units of Z$.$

$[$b$]$ Calculate the TA ratio for each product.$$

$[$c$]$ In the absence of demand restrictions for the three products ,advise XYZ Ltd$\text{'}$s management on optimal production plan.$$

$[$d$]$ Distinguish between throughput accounting and limiting factor analysis.